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Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

About this Title

David Chataur, Lafma, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens Cedex 1, France, Martintxo Saralegi-Aranguren, Laboratoire de Mathématiques de Lens, EA 2462, Université d’Artois, SP18, rue Jean Souvraz, 62307 Lens Cedex, France and Daniel Tanré, Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1214
ISBNs: 978-1-4704-2887-7 (print); 978-1-4704-4744-1 (online)
DOI: https://doi.org/10.1090/memo/1214
Published electronically: April 16, 2018
Keywords: Intersection homology, Intersection cohomology, Thom-Whitney cohomology, Balanced perverse complex, Sullivan minimal model, Blow-up, Formality, Perverse local systems, Filtered spaces, Stratified spaces, CS sets, Pseudomanifolds, Topological invariance, Isolated singularities, Thom spaces, Nodal hypersurfaces, Morgan’s model for the complement of a divisor
MSC: Primary 55N33, 55P62, 57N80

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Table of Contents

Chapters

• Introduction
• 1. Simplicial blow-up
• 2. Rational algebraic models
• 3. Formality and examples
• A. Topological setting

Abstract

Let $X$ be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of $X$, introduced by M. Goresky and R. MacPherson.

We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when $X$ is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan’s differential forms over the field $\mathbb {Q}$. In a second step, we use these forms to extend Sullivan’s presentation of rational homotopy type to intersection cohomology.

For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded ${\mathbb {Q}}$-algebras (CDGA’s) due to Hovey. We establish also the existence and uniqueness of a positively graded, minimal model of some perverse CDGA’s, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PL-pseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in $\mathbb {C}\textrm {P}(4)$, is intersection-formal.